Spin-orbit effects in the hydrogenic impurity levels of wurtzite semiconductors

TL;DR

This paper calculates how spin-orbit interactions affect hydrogenic impurity energy levels in wurtzite semiconductors, comparing intrinsic and extrinsic effects using perturbation theory and Kane model for specific materials.

Contribution

It introduces a first-order perturbation approach to quantify spin-orbit effects on impurity levels in wurtzite semiconductors, including a renormalized extrinsic interaction from Kane theory.

Findings

Intrinsic and extrinsic spin-orbit effects vary among materials.

The extrinsic Rashba interaction is comparable to the intrinsic Dresselhaus effect.

Quantitative estimates provided for GaN, ZnO, InN, and AlN.

Abstract

The corrections to the $E_2^*$ energy level of hydrogenic impurities in semiconductors with wurtzite crystal structure are calculated using first-order perturbation theory in the envelope-function approximation. We consider the intrinsic (Dresselhaus) spin-orbit effective Hamiltonian in the conduction band and compare its effects to the renormalized extrinsic (Rashba) spin-orbit interaction which is analogous to the spin-orbit interaction in the bare hydrogen atom. In order to evaluate the extrinsic spin-orbit interaction we obtain the renormalized coupling constant $\lambda^*$ for wurtzite semiconductors from 8-band Kane theory. We apply our theory to four representative binary semiconductors with wurtzite crystal structure, namely, GaN, ZnO, InN and AlN, and discuss the relative strength of the effects of the intrinsic and extrinsic spin-orbit contributions.

Tables

Table 1. Table 1: Intrinsic spin-orbit interaction corrections to the E 2 ∗ superscript subscript 𝐸 2 E_{2}^{*} energy level of hydrogenic donor impurities in various semiconductors with wurtzite crystal structure. The parameters γ 𝛾 \gamma and b 𝑏 b are also indicated. The effective masses and dielectric constants used to calculate E 2 ∗ superscript subscript 𝐸 2 E_{2}^{*} are given in Table II.

	$\gamma$	$b$	${\epsilon }_{1,2}$	${\epsilon }_{3,4}$	$E_2^{\ast }$	$2A/E_2^{\ast }$	$2B/E_2^{\ast }$
	[${\text{meV}\text{Å}}^3$]		[ $\mu$eV ]	[ $\mu$eV ]	[meV]	$[\%]$	$[\%]$
GaN	400	3.954	$\pm$ 13.24	$\pm$ 52.79	11.97	0.22	0.88
ZnO	320	3.855	$\pm$ 14.67	$\pm$ 58.39	14.65	0.20	0.80
InN	345	4.885	$\pm$ 14.71	$\pm$ 59.49	16.20	0.18	0.73
AlN	6.45	3.767	$\pm$ 3.98	$\pm$ 15.81	70.86	0.011	0.15

Table 2. Table 2: Coupling constant of the effective extrinsic spin-orbit interaction and auxiliary material parameters han for wurtzite semiconductors.

	$m^{\ast }/m_0$	$ϵ$	$E_g$	${\Delta }_0$	${\lambda }_w^{\ast }$
			[eV]	[meV]	[${10}^{-2}$ Å2]
GaN	0.32	9.53	3.51	72.9	5.95
ZnO	0.32	8.62	3.44	43	3.08
InN	0.26	7.39	0.78	40	1.33
AlN	0.38	4.27	5.4	-58.5	-1.04

Table 3. Table 3: Extrinsic spin-orbit corrections to the 2 ​ p 2 𝑝 2p energy level of hydrogenic donor impurities for four important wurtzite semiconductores, along with relevant material parameters.

	$a^{\ast }$	${\lambda }_w^{\ast }$	${\text{Ry}}^{\ast }$	${\epsilon }_1$	${\epsilon }_2$	$\left({\epsilon }_2-{\epsilon }_1\right)/E_2^{\ast }$
	[Å]	[${10}^{-2}$ Å2]	[meV]	[$\mu$eV]	[$\mu$eV]	$[\%]$
GaN	15.8	5.95	11.97	-1.94	0.968	0.024
ZnO	14.1	3.08	14.65	-1.56	0.779	0.016
InN	15.2	1.33	16.20	-0.625	0.313	0.0058
AlN	5.9	-1.04	70.85	14.50	-7.25	-0.031

Table 4. Table 4: Energy corrections due to the combined intrinsic and extrinsic spin-orbit interactions to the energy level E 2 ∗ superscript subscript 𝐸 2 E_{2}^{*} of hydrogenic donor impurities for four important binary semiconductors with wurtzite crystal structure.

	${\epsilon }_{3,4}$	${\epsilon }_5$	${\epsilon }_6$	${\epsilon }_7$	${\epsilon }_8$
	[$\mu$eV]	[$\mu$eV]	[$\mu$eV]	[$\mu$eV]	[$\mu$eV]
GaN	0.23	-13.36	13.13	-54.71	54.48
ZnO	4.90	-49.39	44.94	-60.78	56.33
InN	0.76	-15.42	14.66	-61.31	60.63
AlN	-20.15	-20.44	40.58	-17.56	37.70

Equations

$$H=H_{\text{0}}+H_{\text{int}}+H_{\text{ext}},$$

$$H_0=\frac{p^2}{2m^{\ast }}+V(\mathbf{r}).$$

$$H_{\text{int}}=\alpha ({\sigma }_x{k}_y-{\sigma }_y{k}_x)+\gamma (b{k}_z^2-k_{\parallel }^2)({\sigma }_x{k}_y-{\sigma }_y{k}_x),$$

$${\psi }_{200\eta }={\left(\frac{1}{32\pi a^{\ast 3}}\right)}^{\frac{1}{2}}\left(2-\frac{r}{a^{\ast }}\right)e^{-r/2a^{\ast }}|\eta ⟩$$

$${\psi }_{211\eta }={\left(\frac{1}{64\pi a^{\ast 3}}\right)}^{\frac{1}{2}}\frac{(x+iy)}{a^{\ast }}{e}^{-r/2a^{\ast }}|\eta ⟩$$

$${\psi }_{210\eta }={\left(\frac{1}{32\pi a^{\ast 3}}\right)}^{\frac{1}{2}}\frac{z}{a^{\ast }}{e}^{-r/2a^{\ast }}|\eta ⟩$$

$${\psi }_{21-1\eta }=-{\left(\frac{1}{64\pi a^{\ast 3}}\right)}^{\frac{1}{2}}\frac{(x-iy)}{a^{\ast }}{e}^{-r/2a^{\ast }}|\eta ⟩,$$

$${\overline{\stackrel{ˉ}{H}}}_{\text{int}}=\left(\begin{array}{cccccccc}0& 0& 0& A& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& B& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ A& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& B& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),$$

$$A=\frac{\gamma }{32\sqrt{2}{a}^{\ast 3}}\left(\frac{14}{15}b+\frac{133}{60}\right)$$

$$B=\frac{\gamma }{32\sqrt{2}{a}^{\ast 3}}\left(\frac{62}{15}b+\frac{433}{60}\right).$$

$${\epsilon }^4-{\epsilon }^2(A^2+B^2)+(AB{)}^2=0.$$

$$H_{\text{so}}=\lambda \hat{\mathbf{\sigma }}\cdot \mathbf{k}\times \nabla V_0(\mathbf{r}).$$

$$H_{\text{ext}}={\lambda }^{\ast }\hat{\mathbf{\sigma }}\cdot \mathbf{k}\times \nabla V(\mathbf{r}),$$

$$\mathcal{H}={\mathcal{H}}_0+{\mathcal{H}}_{\mathbf{k}\cdot \mathbf{p}}+{\mathcal{H}}_{\text{so}}$$

$${\mathcal{H}}_0$$

$${\mathcal{H}}_{\mathbf{k}\cdot \mathbf{p}}$$

$${\mathcal{H}}_{\text{so}}$$

$$v_1=\text{arrowvert}iS\uparrow ⟩$$

$$v_2=\text{arrowvert}iS\downarrow ⟩$$

$$\displaystyle v_{3}=-\frac{1}{\sqrt{2}}\left\arrowvert(X+iY)\uparrow\right\rangle$$

$$\displaystyle v_{4}=-\frac{1}{\sqrt{6}}\Big{[}\arrowvert(X+iY)\downarrow\rangle-2\arrowvert Z\uparrow\rangle\Big{]}$$

$$\displaystyle v_{5}=\frac{1}{\sqrt{6}}\Big{[}\arrowvert(X-iY)\uparrow\rangle+2\arrowvert Z\downarrow\rangle\Big{]}$$

$$v_6=\frac{1}{\sqrt{2}}\text{arrowvert}(X-iY)\downarrow ⟩$$

$$\displaystyle v_{7}=-\frac{1}{\sqrt{3}}\Big{[}\arrowvert(X+iY)\downarrow\rangle+\arrowvert Z\uparrow\rangle\Big{]}$$

$$\displaystyle v_{8}=-\frac{1}{\sqrt{3}}\Big{[}\arrowvert(X-iY)\uparrow\rangle-\arrowvert Z\downarrow\rangle\Big{]}.$$

$$\mathcal{H}=\left(\begin{array}{cccccccc}{E}_c& 0& \frac{-1}{\sqrt{2}}{P}_2{k}_{+}& \sqrt{\frac{2}{3}}{P}_1{k}_z& \frac{1}{\sqrt{6}}{P}_2{k}_{-}& 0& \frac{-1}{\sqrt{3}}{P}_1{k}_z& \frac{-1}{\sqrt{3}}{P}_2{k}_{-}\\ 0& E_c& 0& \frac{-1}{\sqrt{6}}{P}_2{k}_{+}& \sqrt{\frac{2}{3}}{P}_1{k}_z& \frac{1}{\sqrt{2}}{P}_2{k}_{-}& \frac{-1}{\sqrt{3}}{P}_2{k}_{+}& \frac{1}{\sqrt{3}}{P}_1{k}_z\\ \frac{-1}{\sqrt{2}}{P}_2{k}_{-}& 0& E_v& 0& 0& 0& 0& 0\\ \sqrt{\frac{2}{3}}{P}_1{k}_z& \frac{-1}{\sqrt{6}}{P}_2{k}_{-}& 0& E_v& 0& 0& 0& 0\\ \frac{1}{\sqrt{6}}{P}_2{k}_{+}& \sqrt{\frac{2}{3}}{P}_1{k}_z& 0& 0& E_v& 0& 0& \\ 0& \frac{1}{\sqrt{2}}{P}_2{k}_{+}& 0& 0& 0& E_v& 0& 0\\ \frac{-1}{\sqrt{3}}{P}_1{k}_z& \frac{-1}{\sqrt{3}}{P}_2{k}_{-}& 0& 0& 0& 0& E_v-{\Delta }_0& 0\\ \frac{-1}{\sqrt{3}}{P}_2{k}_{+}& \frac{1}{\sqrt{3}}{P}_1{k}_z& 0& 0& 0& 0& 0& E_v-{\Delta }_0\end{array}\right)$$

$$\frac{\hslash }{m}⟨-iS\downarrow \text{arrowvert}\mathbf{k}\cdot \mathbf{p}\text{arrowvert}Z\downarrow ⟩=-i\frac{\hslash }{m}{k}_z⟨S\text{arrowvert}{p}_z\text{arrowvert}Z⟩\equiv k_z{P}_1,$$

$$\frac{\hslash }{m}⟨-iS\downarrow \text{arrowvert}\mathbf{k}\cdot \mathbf{p}\text{arrowvert}X\downarrow ⟩=-i\frac{\hslash }{m}{k}_x⟨S\text{arrowvert}{p}_x\text{arrowvert}X⟩\equiv k_x{P}_2.$$

$$T_x=\frac{1}{3\sqrt{2}}\left(\begin{array}{cccc}-\sqrt{3}& 0& 1& 0\\ 0& -1& 0& \sqrt{3}\end{array}\right),$$

$$T_y=\frac{-i}{3\sqrt{2}}\left(\begin{array}{cccc}\sqrt{3}& 0& 1& 0\\ 0& 1& 0& \sqrt{3}\end{array}\right),$$

$$T_z=\frac{\sqrt{2}}{3}\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right).$$

$$\left(\begin{array}{ccc}(E_c+V){\mathbb{I}}_{2\times 2}& \sqrt{3}{P}_1\mathbf{T}\cdot {\mathbf{k}}_{\alpha }& \frac{-1}{\sqrt{3}}{P}_1\hat{\mathbf{\sigma }}\cdot {\mathbf{k}}_{\alpha }\\ \sqrt{3}{P}_2{\mathbf{T}}^{†}\cdot {\mathbf{k}}_{\alpha }& (E_v+V){\mathbb{I}}_{4\times 4}& 0\\ \frac{-1}{\sqrt{3}}{P}_1\hat{\mathbf{\sigma }}\cdot {\mathbf{k}}_{\alpha }& 0& (E_v-{\Delta }_0+V){\mathbb{I}}_{2\times 2}\end{array}\right),$$

$$\left[\mathbf{T}\cdot {\mathbf{k}}_{\alpha }\frac{3P_1^2}{E-V+E_g}{\mathbf{T}}^{†}\cdot {\mathbf{k}}_{\alpha }+\hat{\mathbf{\sigma }}\cdot {\mathbf{k}}_{\alpha }\frac{P_1^2}{3(E-V+E_g)}\hat{\mathbf{\sigma }}\cdot {\mathbf{k}}_{\alpha }\right]{\psi }_c=(E-V){\psi }_c.$$

$$H_{\text{ext}}={\lambda }_w^{\ast }\hat{\mathbf{\sigma }}\cdot \left({\mathbf{k}}_{\alpha }\times {\nabla }_{\alpha }V\right).$$

$${\lambda }_w^{\ast }=\frac{ϵP_1^2}{3}\left[\frac{2}{E_g^2}-\frac{1}{(E_g+{\Delta }_0{)}^2}\right],$$

$$H_{\text{ext}}=\frac{24a^{\ast 3}}{r^3}{\xi }_{2p}^{\ast }\mathbf{L}\cdot \mathbf{S},$$

$${\xi }_{2p}^{\ast }\equiv \frac{e^2{\lambda }_w^{\ast }}{24ϵ{\hslash }^2{a}^{\ast 3}}.$$

Full text

I Introduction

Hydrogenic impurities are a central aspect of semiconductor physics and technology.pan ; yu-car In recent years, impurity states have been proposed as potential qubits in quantum information devices.kan ; koi-hu-das In binary III-V and II-VI semiconductors, the spin-orbit interaction can play an important role in the electronic structure of confined electronic states.win ; int-tam-wei At the level of the effective-mass approximation, a hydrogenic donor impurity and the hydrogen atom are almost completely analogous quantum systems.koh Thus, it is in principle a simple matter to describe at that level the effect of the so-called extrinsic or Rashba spin-orbit interaction in the impurity states. This contribution to the spin-orbit interaction in local external potentials in solids is, in this context, analogous to the spin-orbit coupling in the hydrogen atom. However, care must be taken due to the necessary renormalization of the spin-orbit coupling constant and to possible modifications to the spin-orbit formula due to anisotropies of the crystal structure. On the other hand, an additional contribution to the spin-orbit interaction, which is specific to the solid-state context must be taken into consideration. This so-called intrinsic or Dresselhaus contributiondre is present in the bulk of the material and reappears in the envelope-function approximation theory of external, mesoscopic, potentials, like the one caused by a ionized donor.

In this article we calculate the energy levels of the $n=2$ shell of hydrogenic impurities of semiconductors with wurtzite crystal structure in the presence of these two contributions to the spin-orbit interaction. While, as mentioned above, the calculation of the extrinsic contribution is in a sense a straightforward application of the well-known formulas for the hydrogen atom, we need to fill a theoretical gap, caused by the anisotropy of the wurtzite crystal structure. To that effect, we obtain here an expression for the extrinsic spin-orbit interaction in wurtzite semiconductors and the effective spin-orbit coupling constant for conduction band electrons. The calculation of the effects of the intrinsic spin-orbit interaction in this context is entirely new and it is considered acting alone and in combination with the extrinsic contribution.

The article is organized as follows. In Section II we introduce the system and study the effect of the intrinsic spin-orbit interaction. In Section III we derive the effective Hamiltonian of the extrinsic spin-orbit interaction using the Foldy-Wouthuysen transformation adapted to this context and in Section IV we calculate the energy corrections that it produces. In Section V we study the combined effect of both spin-orbit interactions, and in Section VI we provide the concluding remarks.

II Intrinsic spin-orbit interaction in the hydrogenic impurity

We consider an electron bound to a hydrogenic donor impurity in a bulk semiconductor with wurtzite crystal structure. Working at the level of the envelope-function approximation (EFA), both the intrinsic and the extrinsic spin-orbit couplings appear in the Hamiltonian:

[TABLE]

where

[TABLE]

Here $V(\mathbf{r})=-e^{2}/\epsilon r$ is the effective Coulomb potential of the electron bound to the ionized donor impurity. We assume the effective mass $m^{*}$ and the dielectric constant $\epsilon$ to be isotropic, thereby preserving the spherical symmetry of the hydrogenic Hamiltonian $H_{0}$.

The extrinsic spin-orbit coupling, $H_{\text{ext}}$, will be discussed below, and for the moment we focus on the intrinsic contribution, $H_{\text{int}}$, which for semiconductors with wurtzite crystal structure is given by voo-wil-car ; fu-wu ; wan-wu-tsa

[TABLE]

where $k_{\parallel}^{2}=k_{x}^{2}+k_{y}^{2}$ and $\alpha$, $b$, and $\gamma$ are material-dependent parameters which are obtained experimentally or via ab-initio calculations. $\boldsymbol{\hat{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices. While $\alpha$ and $\gamma$ can vary considerably between different materials, $b$ is roughly universal and close to 4 for all materials. Note that $H_{\text{int}}$ has two parts, one of them linear and the other one cubic in the wavevector $k$. The cubic-in-k term displays an anisotropy between the $z$-direction and the directions in the $xy$-plane. This anisotropy and the presence of the linear term distinguish the intrinsic spin-orbit Hamiltonian of wurtzite semiconductors from the Dresselhaus coupling of zincblende semiconductors.dre

The hydrogenic Hamiltonian $H_{0}$ has the renormalized eigenvalues $E_{n}^{*}=-E_{R}^{*}/n^{2}$, where $E_{R}^{*}=m^{\ast}e^{4}/2\epsilon^{2}\hbar^{2}$ is the effective Rydberg energy. The aim of this study is to obtain the corrections to the $E_{2}$ energy level due to the intrinsic and extrinsic spin-orbit Hamiltonians. We will work at the level of first-order perturbation theory, which is adequate due to the smallness of the spin-orbit couplings compared to the separation of the bare $E_{n}^{*}$ levels. In order to diagonalize the intrinsic Hamiltonian $H_{\text{int}}$ in the $E_{2}$ subspace we use the basis of hydrogenic eigenstates of $\{L^{2},L_{z},S^{2},S_{z}\}$, given by

[TABLE]

where $\eta=\{\uparrow,\downarrow\}$. The matrix elements of the linear-in-$k$ terms of $H_{\text{int}}$ in this basis are zero; only the cubic-in-$k$ terms contribute. Ordering the basis states as: $|200\uparrow\rangle$, $|200\downarrow\rangle$, $|211\uparrow\rangle$, $|211\downarrow\rangle$, $|210\uparrow\rangle$, $|210\downarrow\rangle$, $|21,-1\uparrow\rangle$, and $|21,-1\downarrow\rangle$, the matrix of $H_{\text{int}}$ in the $n=2$ subspace is

[TABLE]

where

[TABLE]

The secular equation for $\bar{\bar{H}}_{\text{int}}$, $\text{det}\left(\bar{\bar{H}}_{\text{int}}-\varepsilon\bar{\bar{I}}\right)=0$, yields

[TABLE]

The eigenvalues are then $\varepsilon_{1,2}=\pm A$ and $\varepsilon_{3,4}=\pm B$. The other four eigenvalues are degenerate and equal to zero. In Table I we present the non-zero energy corrections for the materials GaN, ZnO, InN and AlN, along with their $\gamma$ and $b$ parameters. In the last two columns we present the energy splittings $2A$ and $2B$ as percentages of the unperturbed energy $E_{2}^{*}$.

III Derivation of the Extrinsic spin-orbit interaction

The spin-orbit Hamiltonian of an electron in vacuum in the presence of an electrostatic potential $V_{0}(\mathbf{r})$ is given by

[TABLE]

When the electron is immersed in a semiconductor in the presence of a mesoscopic potential $V(\mathbf{r})$, the effective extrinsic spin-orbit Hamiltonian takes the form

[TABLE]

where $\lambda^{\ast}$ is an effective coupling constant. This expression is valid for semiconductors with zincblende crystal structure, which presents a basic cubic symmetry. The wurtzite crystal structure has less symmetry than the zincblende, due to the special role of its c-axis. This lack of isotropy is also present in the intrinsic spin-orbit coupling given above, Eq. (3). In what follows we shall derive an expression analogous to Eq. (9) for semiconductors with wurtzite crystal structure.

We start with the $\mathbf{k}\cdot\mathbf{p}$ crystal Hamiltonian:

[TABLE]

where

[TABLE]

and $U$ is the periodic crystal potential. We will write the matrix of $\mathcal{H}$ in the common basis of $H_{0}$ and $J_{z}$ given by:

[TABLE]

Here $\arrowvert S\eta\rangle$ are conduction-band $s$-states, with energy $E_{c}$, and $\arrowvert X\eta\rangle$, $\arrowvert Y\eta\rangle$, and $\arrowvert Z\eta\rangle$ are valence-band $p$-type states, with energy $E_{v}$. The energy gap is given by $E_{g}=E_{c}-E_{v}$. We calculate the matrix elements $\mathcal{H}_{ij}=\langle v_{i}\,\arrowvert\mathcal{H}\arrowvert v_{j}\,\rangle$, where $\{i,j=1,\ldots,8\}$, and obtain

[TABLE]

where $k_{\pm}=k_{x}\pm ik_{y}$ and $\Delta_{0}=\frac{\hbar}{4m^{2}c^{4}}\langle X|\frac{\partial U}{\partial x}P_{y}-\frac{\partial U}{\partial y}P_{x}|Y\rangle$ is the spin-orbit splitting of the valence bands. We have defined the constants $P_{1}$ and $P_{2}$ coming from the matrix elements:

[TABLE]

[TABLE]

We now introduce the impurity potential $V(\mathbf{r})$, which varies slowly in the length scale of the lattice constant. Its matrix elements in the basis {$v_{i}$} are essentially diagonal thanks to the orthogonality of the basis set and its slow variation in atomic scale. In short, we are applying here the envelope function approximation. The matrix of $\mathcal{H}+V$ can be expressed in a compact form using the matrices $\mathbf{T}$ familiar from group theory:win

[TABLE]

[TABLE]

[TABLE]

Using these matrices, the Hamiltonian matrix becomes:

[TABLE]

where $\mathbf{k}_{\alpha}=(\alpha k_{x},\alpha k_{y},k_{z})$ and $\alpha=P_{2}/P_{1}$.

Following the application of the Foldy-Wouthuysen transformation described by Winkler for zincblende semiconductors,win we obtain an effective equation, restricted to the conduction band, for the electronic states in the donor impurity

[TABLE]

Using the relation $(\boldsymbol{\hat{\sigma}}\cdot\mathbf{A})(\boldsymbol{\hat{\sigma}}\cdot\mathbf{B})=\mathbf{A}\cdot\mathbf{B}+i\boldsymbol{\hat{\sigma}}\cdot\left(\mathbf{A}\times\mathbf{B}\right)$, we obtain two terms from the second term in the above equation, one of which corresponds to the effective spin-orbit interaction in the conduction band:

[TABLE]

where we defined $\boldsymbol{\nabla}_{\alpha}\equiv(\alpha\frac{\partial}{\partial x},\alpha\frac{\partial}{\partial y},\frac{\partial}{\partial z})$. We have identified the coupling constant for the extrinsic spin-orbit interaction in wurtzite semiconductors:

[TABLE]

analogous to the known coupling constant $\lambda^{*}$ in Eq. (9) for zincblende materials.

Note the factor $\alpha\equiv P_{2}/P_{1}$ in Eq. (20), which reflects the anisotropy of the wurtzite crystal structure. The Coulomb potential of the hydrogenic impurity, $V(\mathbf{r})$, that appears in Eq. (20), was introduced in Eq. (2). Actually, the spherically symmetric form given after Eq. (2) is a simplified expression which does not include the effect of the anisotropic effective mass and dielectric constant of wurtzite crystal structures.han As a first approximation, here we will work with this spherically symmetric Coulomb potential and will also disregard the $\alpha$-dependence of $\boldsymbol{\nabla}_{\alpha}$ and $\mathbf{k_{\alpha}}$. A complete treatment of the anisotropy effects would require considering the modified eigenvalue problem of the anisotropic hydrogenic impurity, and then the effect of the factor $\alpha$ in the spin-orbit interaction. We leave this refined treatment for future work. In Table II we present the values of $\lambda^{*}_{w}$ for GaN, ZnO, InN, and AlN, along with the material parameters needed to evaluate Eq. (21).

IV Extrinsic spin-orbit corrections to the $2p$ level of hydrogenic impurities

Using Eqs. (20) and (21) we obtain the Hamiltonian of the extrinsic spin-orbit interaction due to the Coulomb potential of the hydrogenic the donor impurity:

[TABLE]

where

[TABLE]

As anticipated above, we simplified the Hamiltonian $H_{\text{ext}}$ by setting the ratio $\alpha=1$ in Eq. (20). We thus revert to the standard spin-orbit coupling of the hydrogen atom, but take into account the appropriate, renormalized, coupling constant $\lambda^{*}_{w}$. The calculation of the first-order corrections to the $2p$ energy level of the impurity due to $H_{\text{ext}}$ now follows the standard treatment of spin-orbit interaction in the hydrogen atom. The common eigenvalues of $H_{\text{ext}}$ and $J^{2}$ are given by:

[TABLE]

for $j=1/2$, and

[TABLE]

for $j=3/2$. We thus obtain for the energy corrections:

[TABLE]

The numerical values of $\varepsilon_{1}$ and $\varepsilon_{2}$ are shown in Table III, together with the parameters needed for their evaluation. We also give the energy variation as a percentage of the unperturbed energy, $(\varepsilon_{2}-\varepsilon_{1})/E_{2}^{\ast}$. One can see that the splitting due to extrinsic spin-orbit interaction is four orders of magnitude smaller than the energy of the original level. This ratio is small but it is not negligible as it is, in fact, one order of magnitude larger than the one obtained for the hydrogen atom, which is equal to 0.00133 %. It should be emphasized that this comparison between the hydrogenic impurity and the hydrogen atom was not obvious a priori, since the renormalization of the coupling constant $\lambda$ is very pronounced (6 orders of magnitude) and could have produced results radically different.

We remark that for aluminum nitride (AlN) the relation between $\varepsilon_{1}$ and $\varepsilon_{2}$ is inverted. This peculiarity originates in the particular characteristics of its electronic structure, which cause $\lambda^{\ast}$ to become negative.

V Combined intrinsic and extrinsic spin-orbit interactions

We now study the effects of the intrinsic and extrinsic spin-orbit interactions combined on the $E_{2}^{*}$ energy level of hydrogenic impurities in wurtzite semiconductors. Thus, we now consider the complete Hamiltonian, Eq. (1), with $H_{\text{int}}$ and $H_{\text{ext}}$ given in Eqs. (3) and Eq. (22), respectively. We will perform again a first-order perturbative treatment, now considering the full spin-orbit Hamiltonian $H_{\text{int}}+H_{\text{ext}}$ as the perturbation. To that end, we will express the spin-orbit coupling in the so-called uncoupled basis states of the $E_{2}^{*}$ shell, that is, $\left\{|l,s;m_{l},m_{s}\rangle\right\}$, used in Section II to treat the intrinsic-alone case. The matrix of the combined spin-orbit Hamiltonian in this basis is:

[TABLE]

where $\beta$ was defined in Eq. (26) and $A$ and $B$ have been defined in Eqs. (6).

The characteristic polynomial that solves the eigenvalue problem is:

[TABLE]

and the corresponding eigenvalues are:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We remark that the energy corrections given in Eqs. (29)–(32) contain the previous cases (intrinsic and extrinsic spin-orbit interactions acting alone) in the appropriate limits, and they are represented schematically in Fig. 1. This plot offers a qualitative view of the energy splittings and shows the greater breaking of degeneracy caused by the combined action of both spin-orbit couplings. Finally, we calculate the eigenvalues $\varepsilon_{i}$ ($i=3,\ldots,8$) using the values of $A$, $B$ y $\beta$ corresponding to GaN, ZnO, InN y AlN; the results are given in Table IV.

VI Conclusion

We have studied theoretically the effects of the spin-orbit interaction on the $E_{2}^{*}$ energy level of hydrogenic donor impurities embedded in semiconductors with wurtzite crystal structure. Both the intrinsic (Dresselhaus) and extrinsic (Rashba) spin-orbit interations have been considered, first acting separately and then together. The study was carried out at the level of first-order perturbation theory, which turns out to be appropriate given the relative magnitude of the corrections to the unperturbed energy spacings. Furthermore, in order to evaluate the extrinsic spin-orbit interaction it was necessary to calculate the renormalized coupling constant $\lambda^{*}$ for wurtzite semiconductors from 8-band Kane theory.

We applied our calculations to four currently important semiconductors, i.e. GaN, ZnO InN, and AlN. A general conclusion of these calculations is that both spin-orbit couplings produce relative energy corrections that are bigger than the standard spin-orbit corrections to the $E_{2}$ energy level of the hydrogen atom. While for GaN, ZnO, InN we conclude that the intrinsic spin-orbit interaction produces larger energy corrections than the extrinsic one, that is not the case for AlN, where both interactions have comparable effects. Another anomaly shown by AlN is the fact that its effective coupling constant $\lambda^{*}$ is negative. This causes the eigenvalues $\varepsilon_{3,4}$, which are positive for GaN, ZnO and InN, to become negative for AlN. These anomalies of AlN are due to the specific features of its electronic structure which determine the relevant parameters $\Delta_{0}$, $\gamma_{w}$ and $a^{\ast}$. Finally, we have found that the combined action of both types of spin-orbit coupling leads to an almost complete breaking of the degeneracy of the unperturbed energy level.

Acknowledgements.

We thank Rodolfo Jalabert and Dietmar Weinmann for discussions that motivated this work. We gratefully acknowledge financial support from Projects UBACYT and CONICET PIP.